F₀₀₀-related things

For the purposes of calculating a conventional Fourier synthesis, both the presence and the value of F₀₀₀ can safely be ignored. The reason is, of course, that for the conventional syntheses, F₀₀₀ is simply a constant term that is added to the electron density distribution. Changing its value will only change the mean electron density and nothing more. Given that most macromolecular crystallographers prefer to contour their maps with first contour at the mean electron density plus somethingxrmsd, it has become a macromolecular norm to actually prefer setting the F₀₀₀ to zero, so that the first contour of the maps is always at somethingxrmsd.

F=0 is bound to fail with the maxent maps. Let me illustrate this with an example. The following graphs show the distribution of density along a line containing the origin peak of a Patterson function projection¹⁷, both for the conventional synthesis and a number of GraphEnt maps calculated with different values for the F₀₀₀ term (all scaled to 999.0). I'm probably taking the fun out of it, but I think it is worth mentioning that this is a Harker line for a single-site platinum derivative : The signal is the major non-origin peak. The other peaks do not arise from the heavy atom structure.

Figure: Conventional map

Figure: F000=38000e

Figure: F000=5000e

Figure: F000=1000e

Figure: F000=100e

Figure: F000=10e

Taking the trends apparent from these graphs to their extreme, you could argue that as the value of F₀₀₀ tends to 0.0 e^-, the peaks in the map will tend towards functions. This line of reasoning immediately warns you that by ``adjusting'' the value of F₀₀₀, you can make your map look as sharp as you please although your data (meaning the data that you have indeed measured) are the same. The point is of course that F₀₀₀ is NOT an adjustable quantity : the sharpness of these maps is not required by the data that you measured, but by the value that you arbitrarily decided to assign to the F₀₀₀. What GraphEnt will give you is (or, better still, I hope it is) what is required by the data (including the assignment of F₀₀₀). If you tell the program that F₀₀₀ = 10.0 e^-, then GraphEnt will give you peaks as sharp as needed for the sum of electron density on the unit cell to be 10.0 e^-. The result will be that noise will also appear as sharp peaks, and you are bound to mis-interprete your map¹⁸.

The one and only consistent way of doing the calculation is to give F₀₀₀ its correct value. This sounds very nice, but in real life things are not so straightforward : what should the F₀₀₀ value be for an isomorphous difference Patterson calculation using acentric terms (in which case even knowing from before-hand the number of substitution sites doesn't help because F_PH - F_P F_H) ? what should the F₀₀₀ value be for a (2mF_o - DF_c)exp(i) difference map phased from an incomplete poly-alanine model ? should the F₀₀₀ include the number of electrons due to bulk solvent although I only have data from 8Å (and some strong data are missing because they were overloaded) ? etc. For these reasons, and in order to keep the procedure of running GraphEnt automatic (at least for the first time), I have resorted to the following unjustified and arbitrary assumptions about your F₀₀₀s :

• Phased syntheses : Assuming a that your crystals contain 50% 2M ammonium sulphate and 50% protein, their mean electron density is expected to be around 0.40 e^-/ų. The assignment then is F₀₀₀ = 0.40V_cell e^-, where V_cell is the volume of your unit cell in ų. I sincerely hope that for the majority of macromolecular problems this is an overstimate of the true value (which is no harm. The maps will not be as sharp as they ought to, but it will not be possible to mis-interprete them). If on the other hand, you are calculating a Fourier synthesis for the heavy atom structure (in which case the assumed F₀₀₀ is much too high), you are better off stoping the calculation after the MAXENT_AUTO.IN file has been produced, edit it and add a reasonable definition for F000.
• Patterson syntheses : In this case, and because I expect most Patterson calculations to involve macromolecular isomorphous differences, I have resorted to F₀₀₀ = [2max(F)]², where max(F) is the largest amplitude observed. This is a rather dubious choice which will almost certainly fail if you are calculating, for example, a native Patterson function.

A pragmatist's view : If your GraphEnt maps look unjustifiably sharp, increase F₀₀₀. If they look smooth, decrease F₀₀₀ till the point where you can still ``interprete'' the features that you see.

Please note : The value of the F₀₀₀ is only used for the calculation of the initial uniform map, but is not used to constrain the sum of densities in the GraphEnt maps that follow. In other words, do not expect the F₀₀₀ calculated from the GraphEnt map to be identical with the value that you defined.

Quoting from Gull & Daniell, (1978), ``... Exact fitting also implies the existence of numerous separate constraints, resulting mathematically in an unwieldy proliferation of Lagrange multipliers and preventing calculation of the solution in all but the simplest cases''. In the case of F₀₀₀ things may not be that complex (I would think that one additional re-scaling step is all that is required), but given the difficulties with estimating F₀₀₀ in the case of Patterson and difference Fourier synthesis, I thought I would better leave F₀₀₀ unconstrained.

Footnotes

... projection¹⁷

This is the line v = 0.5 from the example Patt_projection.in included with the distribution of GraphEnt.

... map¹⁸

You can actually see one of the artifacts of having too small a value of F₀₀₀ in the last two graphs. If you look carefully, you will see that it is not only the major peak that is beginning to show line splitting, but also the origin peak. The splitting of the origin peak is only indirectly due to the F₀₀₀ being too small : as the peaks in the GraphEnt map tend towards functions, the amplitudes of the transform of the GraphEnt map tend to a set of normalised E-values with = 1 for all resulution shells. Now, because you are sampling data that go to the infinity on a finite grid (ie, the grid of your map), the power of the transform that is outside the limits of your finite grid folds back into the limits of your transform (this is usually called ``aliasing''). The most notable result is that some of the phases of the Patterson function coefficients will become negative, and the origin peak will start developing a hole in the middle.